ABSTRACT ALGEBRACyclic Groups

Republic of the PhilippinesPANGASINAN STATE UNIVERSITYLingayen Campus

OBJECTIVES:Recall the meaning of cyclic groupsDetermine the important characteristics

of cyclic groupsDraw a subgroup lattice of a group

preciselyFind all elements and generators of a

cyclic groupIdentify the relationships among the

various subgroups of a group

Cyclic GroupsThe notion of a “group,” viewed only 30 years ago as the epitome of sophistication, is today one of the mathematical concepts most widely used in physics, chemistry, biochemistry, and mathematics itself.

ALEXEY SOSINSKY , 1991

A Cyclic Group is a group which can be generated by one of its elements.

That is, for some a in G, G={an | n is an element of Z}

Or, in addition notation, G={na |n is an element of Z}

This element a (which need not be unique) is called a generator of

G.Alternatively, we may write G=<a>.

EXAMPLESThe set of integers Z under ordinary addition is

cyclic. Both 1 and −1 are generators. (Recall that, when the operation is addition, 1n is interpreted as

1 + 1 + ∙ ∙ ∙+ 1n terms

when n is positive and as (-1) + (-1) + ∙ ∙ ∙+ (-1)

|n| termswhen n is negative.)

The set Zn = {0, 1, . . . , n−1} for n ≥ 1 is a cyclic group under addition modulo n. Again, 1 and −1 =n−1 are generators.Unlike Z, which has only two generators, Zn may have many generators

(depending on which n we are given).Z8 = <1> = <3> = <5> = <7>.

To verify, for instance, that Z8 = <3>, we note that <3> = {3, 3 + 3, 3 + 3 + 3, . . .} is the set {3, 6, 1, 4, 7, 2, 5, 0} = Z8 . Thus, 3 is a generator of Z8 . On the other hand, 2 is not a generator, since <2>={0, 2, 4, 6} ≠ Z 8.

U(10) = {1, 3, 7, 9} = {30 , 31 , 33 , 32 } = <3>. Also, {1, 3, 7, 9} = {70 , 73 , 71 , 72 } = <7>. So both 3 and 7 are generators for U(10).

Quite often in mathematics, a “nonexample” is as helpful in understanding a concept as an example. With regard to cyclic groups, U(8) serves this purpose; that is, U(8) is not a cyclic group. Note that U(8) = {1, 3, 5, 7}. But

<1> = {1}<3> = {3, 1}<5> = {5, 1}<7> = {7, 1}

so U(8) ≠ <a> for any a in U(8).

With these examples under our belts, we are now ready to tackle cyclic groups in an abstract way and state their key properties.

Properties of Cyclic GroupsTheorem 4.1 Criterion for ai = aj

Let G be a group, and let a belong to G. If a has infinite order, then aia j if and only if i=j. If a has finite order, say, n, then <a>= {e, a, a2, . . . , an–1} and aia j if and only if n divides i – j.

Theorem Every cyclic group is abelian.

PROOF Let G be a cyclic group generated by g . Let a, b be elements of G. We want to show that ab = ba. Now, a = gm and b = gn for some integers a and b. So, ab = gmgn = gm+n and ba = gngm = gn+m. But m+n = n+m (addition of integers is commutative). So ab = ba.

EXAMPLES(i) (Z, +) is a cyclic group because Z = <1>.(ii) ({na | n ∈ Z}, +) is a cyclic group, where a is any fixed element of Z.(iii) (Zn,+n) is a cyclic group because Zn = <[1]> .

Corollary 1 |a| = |<a>|For any group element a, |a| = |<a>|.

Corollary 2 ak =e Implies That |a| Divides kLet G be a group and let a be an element of order n in G. If ak = e, then n divides k.

Theorem 4.1 and its corollaries for the case |a| = 6 are illustrated in Figure 4.1.

Theorem 4.2 <ak> = <a gcd(n,k)>Let a be an element of order n in a

group and let k be a positive integer. Then <ak>=<agcd(n,k)> and |ak| = n/gcd(n,k).

Corollary 1 Orders of Elements in Finite Cyclic Groups

In a finite cyclic group, the order of an element divides the order of the group.

Corollary 2 Criterion for <ai> = <aj > and |ai | = |aj |

Let |a| = n. Then <ai> = <aj > if and only if gcd(n, i) = gcd(n, j) and |ai | = |aj | if and only if gcd(n, i) 5 gcd(n, j) .

Corollary 3 Generators of Finite Cyclic Groups

Let |a| = n. Then <a> = <a j > if and only if

gcd(n, j) = 1 and |a|= |<a j >| if and only if

gcd(n, j) = 1. Corollary 4 Generators of Zn

An integer k in Zn is a generator of Zn

if and only if gcd(n, k) = 1.

Classification of Subgroups of Cyclic GroupsTheorem 4.3 Fundamental Theorem of Cyclic Groups

Every subgroup of a cyclic group is cyclic. Moreover, if |<a>| = n, then the order of any subgroup of <a> is a divisor of n; and, for each positive divisor k of n, the group <a> has exactly one subgroup of order k—namely, <an/k>.

Corollary Subgroups of Zn

For each positive divisor k of n, the set <n/k> is the unique subgroup of Zn of order k; moreover, these are the only subgroups of Zn.

EXAMPLE The list of subgroups of Z30 is<1>= {0, 1, 2, . . . , 29} order 30,<2>= {0, 2, 4, . . . , 28} order 15,<3>= {0, 3, 6, . . . , 27} order 10,<5>= {0, 5, 10, 15, 20, 25}

order 6,<6>= {0, 6, 12, 18, 24} order 5,<10>= {0, 10, 20} order 3,<15>= {0, 15} order 2,<30>= {0} order 1.

Theorem 4.4 Number of Elements of Each Order in a

Cyclic GroupIf d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is φ(d).

Corollary Number of Elements of Order d in a Finite Group

In a finite group, the number of elements of order d is divisible by φ (d).

The lattice diagram for Z30 is shown in Figure 4.2. Notice that <10> is a subgroup of both <2> and <5>, but <6> is not a subgroup of <10>.

PREPATRED BY:TONY A. CERVERA JR.III-BSMATH-PURE